Does Cycles support all (relatively new) CUDA GPUs?
GPU Rendering Page: https://www.blender.org/manual/render/cycles/gpu_rendering.html

Blender supports GPU rendering of CUDA cards with computing power 2.0 and above. (See chart here: https://developer.nvidia.com/cuda-gpus)

So as long as the GPU is on the NVIDIA CUDA chart with a processing power of atleast 2.0, you are good to go. If it not on the chart you are running a gamble whether it will work or not. (Unless of course your friend confirms specfic GPU will work because he has tried it.)

As for what to Specfically look for, I am not exactly the expert but I can say that Blender cannot render a scene IF the size of the scene is larger than the memory of the video card. For example:
A scene that is 2.5 gb will not render on a 2gb video card.
A scene that is 2.5 gb will render on a 3gb video card.

And obviously try to shoot for a higher porcessiung power when buying. Here is a 'blenchmark' sheet for GPUs: http://blenchmark.com/gpu-benchmarks

I personally use the GTX 970.